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April 27th, 2018, 02:03 AM   #1
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limit of arctan(x/m)

please check the given limit is correct or not. as the value of arctanx or tan inverse x cannot exceed π/2, but its limit written here is π for m<O.
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April 27th, 2018, 02:08 AM   #2
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You are right, it doesn't make any sense.
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April 27th, 2018, 03:33 AM   #3
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It depends on the definition of $\arctan x$. If we define $$-\frac\pi2 < \arctan x < \frac\pi2$$
then you are correct, but if we define
$$0 < \arctan x < \pi$$
then the book is correct.
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April 27th, 2018, 06:22 AM   #4
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Does the book allow $s$ to be negative?
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April 27th, 2018, 10:52 PM   #5
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Quote:
Originally Posted by v8archie View Post
It depends on the definition of $\arctan x$. If we define $$-\frac\pi2 < \arctan x < \frac\pi2$$
then you are correct, but if we define
$$0 < \arctan x < \pi$$
then the book is correct.
but, if we define arctanx to (-π,π) ,
will arctanx remain a function?
it will have three values for one x.
Help me if I am wrong.
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April 27th, 2018, 10:56 PM   #6
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Quote:
Originally Posted by Shariq Faraz View Post
please check the given limit is correct or not. as the value of arctanx or tan inverse x cannot exceed π/2, but its limit written here is π for m<O.
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Originally Posted by skipjack View Post
Does the book allow $s$ to be negative?
s can be any complex or real number, but greater than 0.
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April 27th, 2018, 11:01 PM   #7
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Quote:
Originally Posted by Shariq Faraz View Post
but, if we define arctanx to (-π,π) ,
will arctanx remain a function?
it will have three values for one x.
Help me if I am wrong.
Don't define it on $(-\pi,\pi)$ then. I suggest you consider the interval that I mentioned instead.
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April 27th, 2018, 11:18 PM   #8
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If $s$ isn't real, "greater than zero" doesn't make sense.

If $s$ is real and positive, the original question is effectively about a one-sided limit.

It's possible for arctan(0) to be defined as $\pi$ (or some other multiple of $\pi$), but it's usually defined as 0. The situation with the arccot function is rather different... it's much easier to find differences of opinion about arccot(0).
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